{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T18:33:49Z","timestamp":1784140429760,"version":"3.55.0"},"reference-count":55,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,10,29]],"date-time":"2023-10-29T00:00:00Z","timestamp":1698537600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007446","name":"King Khalid University, Abha, Saudi Arabia","doi-asserted-by":"publisher","award":["RGP2\/366\/44"],"award-info":[{"award-number":["RGP2\/366\/44"]}],"id":[{"id":"10.13039\/501100007446","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The purpose of this article is to introduce and study certain families of normalized certain functions with symmetric points connected to Gegenbauer polynomials. Moreover, we determine the upper bounds for the initial Taylor\u2013Maclaurin coefficients |a2| and |a3| and resolve the Fekete\u2013Szeg\u00f6problem for these functions. In addition, we establish links to a few of the earlier discovered outcomes.<\/jats:p>","DOI":"10.3390\/axioms12111018","type":"journal-article","created":{"date-parts":[[2023,10,29]],"date-time":"2023-10-29T05:01:08Z","timestamp":1698555668000},"page":"1018","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Coefficient Bounds and Fekete\u2013Szeg\u00f6 Inequalities for a Two Families of Bi-Univalent Functions Related to Gegenbauer Polynomials"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1149-9559","authenticated-orcid":false,"given":"Yahya","family":"Almalki","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5838-7365","authenticated-orcid":false,"given":"Abbas Kareem","family":"Wanas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Al-Qadisiyah, Al Diwaniyah 58001, Iraq"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5881-9260","authenticated-orcid":false,"given":"Timilehin Gideon","family":"Shaba","sequence":"additional","affiliation":[{"name":"Department of Physical Sciences, Landmark University, Omu-Aran 251103, Nigeria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2855-7535","authenticated-orcid":false,"given":"Alina","family":"Alb Lupa\u015f","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Oradea, Str. Universitatii nr. 1, 410087 Oradea, Romania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0165-4992","authenticated-orcid":false,"given":"Mohamed","family":"Abdalla","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,29]]},"reference":[{"key":"ref_1","unstructured":"Legendre, A. (1785). Recherches sur la Attraction Sph\u00e9roides Homog\u00e9nes, M\u00e9moires Pr\u00e9sentes par Divers Savants a la Acad\u00e9mie des Sciences de la Institut de France."},{"key":"ref_2","unstructured":"Bateman, H. (1953). Higher Transcendental Functions, McGraw-Hill."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Doman, B.G.S. (2015). The Classical Orthogonal Polynomials, World Scientific.","DOI":"10.1142\/9700"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/S0377-0427(02)00642-8","article-title":"The Gegenbauer polynomials and typically real functions","volume":"153","author":"Kiepiela","year":"2003","journal-title":"J. Comput. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1112\/jlms\/s1-8.2.85","article-title":"Eine Bemerkung \u00fcber ungerade schlichte Funktionen","volume":"2","author":"Fekete","year":"1933","journal-title":"J. Lond. Math. Soc."},{"key":"ref_6","unstructured":"Duren, P.L. (1983). Univalent Functions, Springer. Grundlehren der Mathematischen Wissenschaften, Band 259."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1188","DOI":"10.1016\/j.aml.2010.05.009","article-title":"Certain subclasses of analytic and bi-univalent functions","volume":"23","author":"Srivastava","year":"2010","journal-title":"Appl. Math. Lett."},{"key":"ref_8","first-page":"405","article-title":"Coefficient estimates for a subclass of analytic bi-univalent functions","volume":"55","author":"Adegani","year":"2018","journal-title":"Bull. Korean Math. Soc."},{"key":"ref_9","first-page":"10","article-title":"On subclass of analytic bi-close-to-convex functions","volume":"13","author":"Alamoush","year":"2021","journal-title":"Int. J. Open Probl. Complex Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2865","DOI":"10.3906\/mat-1906-56","article-title":"Coefficient estimates for a new subclasses of lambda-pseudo bi-univalent functions with respect to symmetrical points associated with the Horadam Polynomials","volume":"43","author":"Alamoush","year":"2019","journal-title":"Turk. J. Math."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Alazman, I., Alkahtani, B.S.T., and Wani, S.A. (2023). Certain properties of \u0394h multi-variate Hermite polynomials. Symmetry, 15.","DOI":"10.3390\/sym15040839"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1016\/j.aml.2011.09.012","article-title":"Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions","volume":"25","author":"Ali","year":"2012","journal-title":"Appl. Math. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Alkahtani, B.S.T., Alazman, I., and Wani, S.A. (2023). Some families of differential equations associated with multivariate Hermite polynomials. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7050390"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"305","DOI":"10.26637\/mjm502\/008","article-title":"On a new subclass of bi-univalent functions of Sakaguchi type satisfying subordinate conditions","volume":"5","author":"Altinkaya","year":"2017","journal-title":"Malaya J. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"34","DOI":"10.56947\/gjom.v5i3.105","article-title":"On the Chebyshev polynomial coefficient problem of some subclasses of bi-univalent functions","volume":"5","author":"Altinkaya","year":"2017","journal-title":"Gulf J. Math."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Amourah, A., Alsoboh, A., Ogilat, O., Gharib, G.M., Saadeh, R., and Al Soudi, M. (2023). A generalization of Gegenbauer polynomials and bi-univalent functions. Axioms, 12.","DOI":"10.3390\/axioms12020128"},{"key":"ref_17","first-page":"36","article-title":"Fekete-Szeg\u00f6 problem for certain subclass of analytic functions with complex order defined by q-analogue of Ruscheweyh operator","volume":"3","author":"Aouf","year":"2020","journal-title":"Constr. Math. Anal."},{"key":"ref_18","first-page":"139","article-title":"Certain class of bi-Bazilevi\u010d functions with bounded boundary rotation involving Sal\u01cege\u01cen operator","volume":"3","author":"Aouf","year":"2020","journal-title":"Constr. Math. Anal."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"23534","DOI":"10.3934\/math.20231196","article-title":"A new subclass of analytic and bi-univalent functions associated with Legendre polynomials","volume":"8","author":"Badghaish","year":"2023","journal-title":"AIMS Math."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Breaz, D., and Cot\u00eerl\u01ce, L.-I. (2022). The study of the new classes of m-fold symmetric bi-univalent functions. Mathematics, 10.","DOI":"10.3390\/math10010075"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Breaz, D., Murugusundaramoorthy, G., Vijaya, K., and Cot\u00eerl\u01ce, L.-I. (2023). Certain class of bi-univalent functions defined by S\u01cel\u01cegean q-difference operator related with involution numbers. Symmetry, 15.","DOI":"10.3390\/sym15071302"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1567","DOI":"10.2298\/FIL1606567B","article-title":"Faber polynomial coefficient estimates for a subclass of analytic bi-univalent functions","volume":"30","author":"Bulut","year":"2016","journal-title":"Filomat"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Caglar, M., Cotirla, L.-I., and Buyankara, M. (2022). Fekete-Szeg\u00f6 inequalities for a new subclass of bi-univalent functions associated with Gegenbauer polynomials. Symmetry, 14.","DOI":"10.3390\/sym14081572"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"10642","DOI":"10.3934\/math.2021618","article-title":"New classes of analytic and bi-univalent functions","volume":"6","year":"2021","journal-title":"AIMS Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"49","DOI":"10.7153\/jca-02-05","article-title":"Certain subclasses of bi-univalent functions satisfying subordinate conditions","volume":"2","author":"Deniz","year":"2013","journal-title":"J. Class. Anal."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"641","DOI":"10.3906\/mat-1503-58","article-title":"Coefficient bounds for subclasses of m-fold symmetric bi-univalent functions","volume":"40","author":"Eker","year":"2016","journal-title":"Turk. J. Math."},{"key":"ref_27","first-page":"383","article-title":"Coefficient bounds for certain classes of bi-univalent functions","volume":"43","author":"Frasin","year":"2014","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1007\/s13370-023-01051-x","article-title":"Certain subclasses of \u03bb-pseudo bi-univalent functions with respect to symmetric points associated with the Gegenbauer polynomial","volume":"34","author":"Ghazy","year":"2023","journal-title":"Afr. Mat."},{"key":"ref_29","first-page":"70","article-title":"Subclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers","volume":"10","author":"Guney","year":"2018","journal-title":"Acta Univ. Sapient. Math."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Hamzat, J.O., Oluwayemi, M.O., Lupa\u015f, A.A., and Wanas, A.K. (2022). Bi-univalent problems involving generalized multiplier transform with respect to symmetric and conjugate points. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6090483"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1024","DOI":"10.3934\/math.2021061","article-title":"Applications of a certain q-integral operator to the subclasses of analytic and bi-univalent functions","volume":"6","author":"Khan","year":"2021","journal-title":"AIMS Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s13370-017-0535-3","article-title":"Chebyshev polynomial coefficient estimates for a class of analytic bi-univalent functions related to pseudo-starlike functions","volume":"29","author":"Magesh","year":"2018","journal-title":"Afr. Mat."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Murugusundaramoorthy, G., and Bulboac\u01ce, T. (2022). Subclasses of Yamakawa-Type Bi-Starlike Functions Associated with Gegenbauer Polynomials. Axioms, 11.","DOI":"10.3390\/axioms11030092"},{"key":"ref_34","first-page":"5","article-title":"Coefficient bounds for certain suclasses of bi-prestarlike functions associated with the Gegenbauer polynomial","volume":"32","author":"Murugusundaramoorthy","year":"2022","journal-title":"Adv. Stud. Contemp. Math."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Orhan, H., and Cot\u00eerl\u01ce, L.-I. (2022). Fekete-Szeg\u00f6 inequalities for some certain subclass of analytic functions defined with Ruscheweyh derivative operator. Axioms, 11.","DOI":"10.3390\/axioms11100560"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1259","DOI":"10.2298\/FIL1506259O","article-title":"Initial coefficient bounds for a general class of bi-univalent functions","volume":"29","author":"Orhan","year":"2015","journal-title":"Filomat"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"P\u00e1ll-Szab\u00f3, A.O., and Oros, G.I. (2020). Coefficient related studies for new classes of bi-univalent functions. Mathematics, 8.","DOI":"10.3390\/math8071110"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"495","DOI":"10.2989\/16073606.2020.1727581","article-title":"Coefficient estimates for some new classes of bi-Bazilevic functions of Ma-Minda type involving the Salagean integro-differential operator","volume":"44","author":"Wanas","year":"2021","journal-title":"Quaest. Math."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2891","DOI":"10.3906\/mat-1507-100","article-title":"On a new subclass of bi-univalent functions defined by using S\u01cel\u01cegean operator","volume":"42","year":"2018","journal-title":"Turk. J. Math."},{"key":"ref_40","first-page":"1","article-title":"On some subclasses of bi-pseudo-starlike functions defined by S\u01cel\u01cegean differential operator","volume":"8","author":"Shaba","year":"2021","journal-title":"Asia Pac. J. Math."},{"key":"ref_41","first-page":"71","article-title":"Coefficient bounds of m-fold symmetric bi-univalent functions for certain subclasses","volume":"12","author":"Shehab","year":"2021","journal-title":"Int. J. Nonlinear Anal. Appl."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1873","DOI":"10.1007\/s40995-018-0647-0","article-title":"Certain subclasses of bi-univalent functions associated with the Horadam polynomials","volume":"43","author":"Srivastava","year":"2019","journal-title":"Iran. J. Sci. Technol. Trans. A Sci."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1007\/s41980-018-0011-3","article-title":"Faber polynomial coefficient estimates for bi-univalent functions defined by the Tremblay fractional derivative operator","volume":"44","author":"Srivastava","year":"2018","journal-title":"Bull. Iran. Math. Soc."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1007\/s13370-016-0478-0","article-title":"Coefficient estimates for some general subclasses of analytic and bi-univalent functions","volume":"28","author":"Srivastava","year":"2017","journal-title":"Afr. Mat."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"103","DOI":"10.37193\/CJM.2018.01.11","article-title":"The Fekete-Szeg\u00f6 functional for a subclass of analytic functions associated with quasi-subordination","volume":"34","author":"Srivastava","year":"2018","journal-title":"Carpathian J. Math."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Wanas, A.K., and Cot\u00eerl\u0103, L.-I. (2022). Applications of (M \u2212 N)-Lucas polynomials on a certain family of bi-univalent functions. Mathematics, 10.","DOI":"10.3390\/math10040595"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Yousef, F., Frasin, B.A., and Al-Hawary, T. (2018). Fekete-Szeg\u00f6 inequality for analytic and bi-univalent functions subordinate to Chebyshev polynomials. arXiv.","DOI":"10.2298\/FIL1809229Y"},{"key":"ref_48","first-page":"85","article-title":"Some subclasses of close-to-convex functions","volume":"2","author":"Thomas","year":"1987","journal-title":"J. Ramanujan Math. Soc."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Miller, S.S., and Mocanu, P.T. (2000). Differential Subordinations: Theory and Applications, Marcel Dekker Inc.","DOI":"10.1201\/9781482289817"},{"key":"ref_50","unstructured":"Erd\u00e9lyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. (1953). Higher Transcendental Functions, McGraw-Hill."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Stein, E., and Weiss, G. (1971). Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press.","DOI":"10.1515\/9781400883899"},{"key":"ref_52","first-page":"625","article-title":"Gegenbauer polynomials and bi-univalent functions","volume":"10","author":"Amourah","year":"2021","journal-title":"Palest. J. Math."},{"key":"ref_53","unstructured":"Reimer, M. (2012). Multivariate Polynomial Approximation, Birkh\u00e4user."},{"key":"ref_54","first-page":"14","article-title":"Chebyshev polynomial bounded for analytic and bi-univalent functions with respect to symmetric conjugate points","volume":"19","author":"Wanas","year":"2019","journal-title":"Appl. Math. E-Notes"},{"key":"ref_55","first-page":"25","article-title":"Coefficient estimates and Fekete-Szeg\u00f6 inequality for family of bi-univalent functions defined by the second kind Chebyshev polynomial","volume":"13","author":"Wanas","year":"2020","journal-title":"Int. J. Open Probl. Compt. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/11\/1018\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:13:44Z","timestamp":1760130824000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/11\/1018"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,29]]},"references-count":55,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2023,11]]}},"alternative-id":["axioms12111018"],"URL":"https:\/\/doi.org\/10.3390\/axioms12111018","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,29]]}}}